A property o f bounded normal operators in Hilbert space

A R K I V F O R M A T E M A T I K B a n d 4 n r 43

C o m m u n i c a t e d 6 D e c e m b e r 1961 b y O. FI~OSTMAlq

A property o f bounded normal operators in Hilbert space

B y GORAN BJORCK a n d YIDAR THOMI~E

Let H be a complex Hilbert space (i.e. a complete inner product space over the complex numbers) with scalar product ( . , . ), norm I1" II, and identity ope- rator I. For any bounded linear operator B in H we denote the spectrum b y sp (B) and the point-spectrum by psp (B). The smallest circular disc containing sp (B) is denoted by dB, the boundary of dB b y cB, and its center and radius b y zB and RB, respectively. For a plane set S we denote the convex hull by cony (S).

For u and v in H we define

sin (u, v) = (1 - ](u, v)]2 ~ 89

\

(If u or v is zero, we define sin (u,v)=O.)

The purpose of this note is to prove and discuss the following Theorem 1. Let N be a bounded normal operator in H. Then

sup/11Null2 I(Nu'-~)12~ = R ~ ,

=+o ~ Ilu[I 2 Ilnll ~ ! (1)

or equivalently, SUPu.0 ~ sin (u, Nu) = RN. (2)

Pro@ We may evidently suppose t h a t RN> 0. For any complex ~t and a n y u E H with I l u l l = l ,

IlNull 2 - I ( N u , u)I s : II N ~ - ~ull 2 - I ( U u ,

(3)

Hence, taking 2 = zm

sup/!lNull~ I<~u-'~u)l~X = sup <

2 - 1 ( N u , ~)-~NI:)

=+o \ l l u l l ~ Ilull 4 ! u =1

< sup

r

Nu]]=l

since the norm and the spectral radius of the normal operator N - zN I are equal.

We will now exhibit a sequence

unEH

and

Ilu.l[+l, ]

II Nu. - u. II +R , when

( N u . , u.) ~ zN

n-+ ~ . (5)

The theorem will then be proved.

We first estabhsh t h a t zNE conv (sp (N)N cN). Otherwise, we could find a clo- sed half-circle

h~CN

Thus there exist two or three distinct complex numbers ~t 1 .. . . . ~trE sp (N) such t h a t ( r = 2 or 3)

12i - ZNr 12 = R2N' r }

ZN=~m~2~

i = l i=1

(6)

r <0~+ ( i = l , r) i n H w i t h B y the properties of spectrum there exist sequences

.tun fn:l

liu )l I = l, (7)

IINu2>-: o1,,= +11+0 when

(u~ >,u~))-->0 for i#?" when n - - > ~ . (9) Finally we define u, = Z [ : I m~ 89 u~ ). The verification of (5) is immediate, using (6),

(7), (8) and (9). This completes the proof.

The proof can, of course, also be carried through b y means of the spectral re- presentation of N (for notation, cf. [2]),

N = f z E ( d x d y ) , z

= x + iy.

I n this case the problem reduces to finding sup

I(da),

a n d ~/~ is the set of positive measures on sp (N) of the form

da=(E(dxdy)u,u)

with

]lull

2I(da) =

f f l z - ~ ] 2 da(z) da($).

One of the authors ([1], Theorems 5 and 6) has studied the problem of maxi-

ARKIV FOR MATEMATIK. B d 4 n r 4 3

mizing this integral for all positive measures on sp (N) with total mass 1, using essentially the above argument.

I t follows from the proof t h a t the sup in (1) and (2) is attained if and only if we can find u E H , I l u l l = l , such t h a t

[[ N u - ZN U 112 = f [ Z -- ZNI 2 (E(dxdy) u, u) = R~N,

(Nu, u) = f z(E(dxdy) u, u) = za.

These equations express t h a t ( E ( d x d y ) u , u ) has its s u p p o r t on cN and its center of g r a v i t y in za. I n particular, the sup is attained if the points ~t in (6) and (8) can be chosen as eigenvalues (and the sequences u~ ) as constant sequences of ei- genvectors), i.e. if ZN E cony (psi) (N) n ca). This situation occurs for instance if sp ( N ) = psp (N), which is, of course, always the case if H is finite-dimensional.

If N is self-adjoint, so t h a t sp (N) is a subset of the real line, then RN is half the diameter of the spectrum, and the sup in (1) and (2) can be attained only in the case discussed above.

I t is easy to construct an example of a (necessarily non self-adjoint) normal operator N in a (necessarily infinite-dimensional) Hilbert space H such t h a t psp (N) is e m p t y and still the sup in (1) and (2) is attained. (One can, for example, arrange t h a t sp (/V)=ca.)

We will a p p l y Theorem 1 and its proof to the following problem: Given u , / E H, w h a t can be said a b o u t the spectrum of a n y bounded normal operator with N u = ] ? The following result follows at once from (4).

Corollary 1. Let u , / e H , Ilull = 1, and let r = II/11 sin (/,u). Let N be a bounded nor- mal operator such that N u = / . Then

o 2

r" ~< R a - I ( / , u) - ZN[ 2. (10) I n particular RN >~ r.

Corollary 1 has the following consequence for self-adjoint operators:

Corollary 2. Let u , / e H , I[ul] = l, r=[[/] I sin (/,u), and let (/,u) be real. Let A be a sel]-ad]oint operator such that A u = ]. Then conv(sp(A)) contains some closed interval [(/, u ) - :~, (/, u ) + flJ w i t h ~ = r ~

Corollary 2 is immediately obtained from Corollary 1. I n fact, conv(sp(A))

= [zA- RA, z~ +RA] and we have to prove t h a t

(ZA + RA -- (/, U)) ((/, U) -- ZA § RA) >t r 2, which in this case is equivalent to (10).

As a trivial consequence of Corollary 2 we get:

Corollary 3. Let u , / e H, HuH = 1, r =

II/Jl

s u p 2 / > (1, u) + r2/(l, ^{u ) =}

II 1113/(/,

). G s p ( A )

This estimate, which can also be easily o b t a i n e d directly, is stronger t h a n t h e obvious ones, (/,u) a n d ]]/]]. I n fact, we h a v e b y the C a u c h y inequality,

(/,u)

II/11

II/llV(/,u).

W e conclude this p a p e r b y discussing t h e following problem: G i v e n u , / E H with I] u]l = 1, can we always find a n o r m a l o p e r a t o r N w i t h N u = / s u c h t h a t we get e q u a l i t y in (10)? T h e answer is contained in the following

Theorem 2. Let u, I E H ,

Ilull

II111

r3 = R2 - I(1, u) - zt 3. ( 1 1 )

Then there exists a normal operator N with N u = / such that RN = R and ZN= Z. I n particular, /or given u , / E H, I] u II = 1, it is always possible t o / l u g a normal operator N such that N u = / and RN = r. I n this case, necessarily zN = z = (/, u).

Proo/. W e first p r o v e the t h e o r e m for the case z = 0 a n d (/,u) real. Consider t h e subspace G in H , s p a n n e d b y u a n d /. W e will p r o v e t h a t we can find t w o orthogonal vectors Y~I a n d 92 such t h a t

I n fact, (12) is solved b y

u=~01+~2' 1 (12)

/ = Ry~ 1 - RyJ2. j

Using the definition of r, we get f r o m (11)

T h e linear o p e r a t o r A defined b y

A ( X l ~ l + x 2 ~ 3 § ) = x l R ~ l - x 2 R y ) 3 , v_L G,

t h e n satisfies the conditions in T h e o r e m 2. T h e o p e r a t o r A is even self-adjoint.

T h e general case can now be reduced to the case considered above. P u t

~/'

ARKIV FOR MATEMATIK. B d 4 n r 4 3

(/',

is real. W e e a s i l y f i n d (cf. (3)) t h a t

r ' = II/']1 sin ( u , l ' ) = Ilill sin ( u , / ) = r,

a n d t h e r e f o r e r '2 = R 2 - (/', u) 2.

W e can t h u s f i n d a s e l f - a d j o i n t o p e r a t o r A w i t h c o n v ( s p ( A ) ) e q u a l t o t h e closed i n t e r v a l [ - R, R] a n d such t h a t A u = ~'. T h e n o r m a l o p e r a t o r N = e -~ ~ A + z I t h e n satisfies t h e c o n d i t i o n s in T h e o r e m 2.

T h e a u t h o r s a r e i n d e b t e d t o P r o f e s s o r L a r s H S r m a n d e r for s t i m u l a t i n g discus- sions on t h e c o n t e n t s of t h i s p a p e r .

University o/Stockholm, Sweden.

R E F E R E N C E S

1. BJORCK, G., Distributions of positive mass, which maximize a certain generalized energy integral. Ark. Mat. 3, 255-269 (1955).

2. RIESZ, F., and Sz.-NAGY, B., Legons d'analyse fonctionelle. Budapest, 1952.

Tryckt den 24 juli 1962

Uppsala 1962. Almqvist & Wiksells Boktryckeri AB